Quantitative particle approximation of nonlinear stochastic Fokker-Planck equations with singular kernel

TL;DR

This paper establishes quantitative convergence of particle systems with singular interactions to solutions of nonlinear stochastic Fokker-Planck equations, including existence and uniqueness results.

Contribution

It introduces a novel approach for analyzing particle approximations of nonlinear stochastic PDEs with singular kernels, proving convergence and well-posedness.

Findings

Empirical measures converge to the nonlinear stochastic Fokker-Planck solution.

Unique strong solutions exist for the considered equations.

Method applies to both repulsive and attractive kernels.

Abstract

We derive quantitative estimates for large stochastic systems of interacting particles perturbed by both idiosyncratic and environmental noises, as well as singular kernels. We prove that the (mollified) empirical process converges to the solution of the nonlinear stochastic Fokker-Planck equation. The proof is based on It\^o's formula for $H_{q}^{1}$-valued process, commutator estimates, and some estimations for the regularization of the empirical measure. Moreover, we show that the aforementioned equation admits a unique strong solution in the probabilistic sense. The approach applies to repulsive and attractive kernels.